Abstract: Ultrasound pulses are emitted by two emitters (12, 14) carried by an instrument (10) and substantially aligned with the tip of the instrument. Several receivers (62a, . . . , 62e) occupy determined positions with respect to the support (50) on which a trace is made with the tip of the instrument so that ultrasound pulses emitted from any point on the surface of the support are received by more than two receivers. The times of propagation of the ultrasound pulses between each of the emitters and receivers are measured in order to evaluate the coordinates of the vector defined by the positions of the two emitters and to deduce therefrom the position of the tip of the instrument on the support, and at least one of the two instrument orientation cues consisting of the inclination of the longitudinal direction of the instrument with respect to the support and the angular position of the instrument about an axis making a predetermined angle with respect to the support.